| Summary: | The thesis considers analytical approaches to the problem of initiation of excitation waves. An excitation wave is a threshold phenomenon. If the initial perturbation is below the threshold, it decays; if it is large enough, it triggers propagation of a wave, and then the parameters of the generated \\rave do not depend on the details of the initial conditions. The problem of initiation of excitation waves is by necessity nonlinear, non-stationary and spatially extended with at least one spatial dimension. These factors make the problem very complicated. There are no known exac...! analytical, or even good asymptotic solutions to this kind of problem in any model, and the practical studies rely on numerical simulations. In this thesis, we develop approaches to this problem based on some asymptotic ideas, but applied in the situation where the 'small parameters' of those methods are not very small. Although results obtained by such methods are not v.ery accurate, they still can be useful if they give qualitatively correct answers in a compact analytical form; such answers can give analytical insights which are impossible or very difficult to gain from numerical simulations. We develop the approaches using, as examples, two simplified models describing fast stages of excitation process: • Zeldovich-Frank-Kamenetskii (ZFK) equation, which is the fast (activator) subsystem of the FitzHugh-Nagumo (FHN) 'base model' of excitable media, and • Biktashev (2002) [8] front model, which is a caricature simplification of the fast subsystem of a typical detailed ionic model of cardiac excitation waves. For these models, we consider two different approaches: • Galerkin-style approximation, where the solution is sought for in a pre-determined analytical form ('ansatz') depending on a few parameters, and then the evolution equation for these parameters are obtained by minimizing the norm of a residual of the partial differential equation (PDE) system, • linearization of the threshold hyper-surface in the functional space, described via linearization of the PDE system on an appropriately chosen solution on that surface (a 'critical solution').
|
|---|
